Wednesday, 30 November 2011

The Jerusalem Square is the "shadow" of the Jerusalem Cube. The ratio between the sizes of a piece of the shape and its copies is an "irrational" number, 1 : ( 1 + root 2 ), or ~2.414213562... .
The square root of two (1.4142...) is the diagonal corner-to-corner distance across a square that has sides of length "one", so if you take a square and butt it up against a copy rotated though 45 degrees, you get the length of the side of the next size up.

As with Fibonacci packings, if you start with a proportion that's way off (say, you try to build the shape with two initial sets of squares of ratio 1:2), then the shape converges on the correct ratio by itself as you add more iterations.

The shield is notable in heraldry for its "illegal" use of precious metal-on-metal colours (gold on silver), which is said to symbolise the idea that the Knights considered themselves above conventional laws. However, it may also be a reference to the quantity of precious metals that went into the area during the period, and which the Knights Templar then hauled away with them when they scarpered.